Theory of highly excited semiconductor nanostructures including Auger coupling: exciton-bi-exciton mixing in CdSe nanocrystals

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Theory of highly excited semiconductor nanostructures including Auger coupling:
exciton-bi-exciton mixing in CdSe nanocrystals
Marek Korkusinski,1 Oleksandr Voznyy,1 and Pawel Hawrylak1
1Quantum Theory Group, Institute for Microstructural Sciences,
National Research Council, Ottawa, Canada, K1A0R6
We present a theory of highly excited interacting carriers confined in a semiconductor nanos-
tructure, incorporating Auger coupling between excited states with different number of excitations.
The Coulomb matrix elements connecting exciton, bi-exciton and tri-exciton complexes are derived
and an intuitive picture of breaking neutral multi-exction complexes into positively and negatively
charged multi-exciton complexes is given. The general approach is illustrated by analyzing the
coupling of biexciton and exciton in CdSe spherical nanocrystals. The electron and hole states
are computed using atomistic sp3d5s∗ tight binding Hamiltonian including an effective crystal field
splitting and surface passivation. For each number of electron-hole pairs the many-body spectrum
is computed in the configuration-interaction approach. The low-energy correlated biexciton levels
are broken into charged complexes: a hole and a negatively charged trion and an electron and a
positively charged trion. Out of a highly excited exciton spectrum a subspace coupled to bi-exciton
levels via Auger processes is identified. The interaction between correlated bi-exciton and exciton
states is treated using exact diagonalization techniques. This allows to extract the spectral function
of the biexciton and relate its characteristic width and amplitude to the characteristic amplitude
and timescale of the coherent time evolution of the coupled system. It is shown that this process
can be described by the Fermi’s Golden Rule only if a fast relaxation of the excitonic subsystem is
accounted for.
I. INTRODUCTION
There is currently renewed interest in the understand-
ing of multi-exciton complexes in highly excited semicon-
ductor nanostructures and their interaction with light.
Two examples are bi-exciton-exciton cascade for the
generation of entangled photon pairs in self-assembled
quantum dots1–5 and enhancing the efficiency of pho-
tovoltaic cells by generation of multi-exciton complexes
(MEG) from a single, high-energy photon absorbed
in semiconductor nanocrystals (NCs).6–12 While much
progress has been achieved in the understanding of multi-
exciton complexes in self-assembled quantum dots13–15
and nanocrystals,16–32 the mixing and decay of complexes
with different numbers of excitons are much less under-
stood.
These processes are important for the understanding
of decoherence in entangled photon pairs and energy-to-
charge conversion. Nozik6 has put forward a proposal
of converting the energy of the high-energy exciton gen-
erated by a high energy photon into several low-energy
electron-hole pairs rather than allowing this energy to be
dissipated. The theoretical threshold photon energy, at
which the excited exciton is expected to convert into a
biexciton, depends on the nanocrystal size, but is typi-
cally about twice the semiconductor bandgap, 2Eg. Such
carrier multiplication process has been demonstrated in
PbSe, PbS, PbTe, CdSe, InAs, and Si NCs,33,34 with ef-
ficiency reaching 700% (seven electron-hole pairs out of
one photon).35 However, a careful analysis of these exper-
iments revised these efficiencies to lower values.10,23,36
Coherent coupling of bi-exciton to highly excited ex-
citon can lead to coherent conversion from biexciton to
exciton and vice versa. However, phonon-assisted relax-
ation in the exciton subsystem was suggested to destroy
coherence and leads to finite bi-exciton lifetime.37 Exact
theoretical estimation of this lifetime is challenging, as
even a realistic computation of single-particle states in
NCs is difficult: for example, for a CdSe NC with diame-
ter of 2.5 nm with 304 atoms, in the energy range of 3Eg
there are approximately 1.58 ·105 exciton and 6.2 ·109 bi-
exciton states. As an approximation, one typically starts
with the computation of NC single-particle states using
the k·p24,25, tight-binding26–29 or empirical pseudopoten-
tial methods.30–32 Next, one builds the electron-hole pair
configurations to be coupled, e.g., the exciton (X) and bi-
exciton (XX) of similar energy, and neglects all Coulomb
coupling (correlations) within each subsystem. Finally,
one computes the Coulomb coupling matrix element be-
tween the chosen X and XX configurations and computes
the lifetime of the XX state using the Fermi’s Golden
Rule.7,9,38–41 As typically there are many X states close
in energy to the XX state, one utilizes a quasi-resonant
approach, scaling the XX-X transition rate by the density
of X states and/or using a “resonance window”.7,9,40–42
A more advanced approach was presented recently by
Witzel et al.43 In this work, a time-dependent evolu-
tion of a single-photon excitation coupled coherently with
multi-exciton states, considered in the k · p approach, is
simulated. Decay of each multiexciton state is accounted
for in the relaxation-time approximation. It is shown
that the relaxation times of different multiexciton com-
plexes, and in particular that of X, play a crucial role in
the efficiency of MEG.
Engineering of materials and nanostructures directed
at optimization of the MEG gain requires therefore a
comprehensive, microscopic theory of (i) coupling of mul-
tiexciton states with different number of excitations, and

2(ii) relaxation processes of these multiexciton states. In
this work we focus on the former, as the latter requires an
additional realistic simulation of the phonon modes in the
NC and a treatment of the carrier-phonon coupling.44,45
We provide a detailed derivation of the Coulomb ma-
trix element coupling the multi-electron-hole configura-
tions differing by one electron-hole pair.46 Further, we
express the states of the system in the basis of configu-
rations with different number of excitations. The form
of this expansion is obtained by exact diagonalization of
the Hamiltonian accounting for all Coulomb interactions
of quasiparticles. From such an eigenstate we extract
the spectral function of the state with the larger number
of pair excitations, assuming that it is “immersed” in a
dense spectrum of the states with fewer pairs. To make
contact with the language of the state lifetime, used in
experiments, we relate the amplitude and the character-
istic width of this spectral function to the amplitude and
time constants of the coherent time evolution of the sys-
tem.
We illustrate this framework on the mixing of X and
XX in a spherical CdSe nanocrystal. The current work is
built upon our recent calculation of the electronic and
optical properties of low energy X and XX states in
the CdSe NC.47 We utilize the QNANO computational
platform48 to carry out atomistic tight-binding computa-
tion of the single-particle states in the NC with diameter
of 3.8 nm. These states are used to construct the cor-
related ground state of the XX as well as the excited
states X∗ with energies close to that of the XX. From
the exact diagonalization of the XX interacting with X∗
in this energy range we extract the spectral function of
the XX ground state and discuss its properties in con-
nection with the coherent time evolution of the coupled
system without relaxation. For the NC studied we find
that the XX-X∗ coupling is weak, resulting in quantum
beats between the XX and X∗ states. We demonstrate
that contact with the description of the XX population
in terms of lifetime can be established only if very fast
decay of the X∗ states due to phonons is assumed.
The paper is organized as follows. In Section II we
present a framework theoretical approach to the system
in which the multiexciton states differing in the number
of excitations are coupled. We establish the general form
of the Hamiltonian for the system, and, on an example
of the X−XX−XXX system, demonstrate the deriva-
tion of the coupling Coulomb matrix elements. We write
the eigenstates of the coupled system, define the spectral
function of the system of n electron-hole pairs immersed
in the spectrum of n− 1 and n+ 1-pair excitations, and
relate this function to the time evolution of the system.
We illustrate these concepts in detail in Section III on
the example of the coupled X − XX system confined
in a single CdSe nanocrystal. In Section IV we present
conclusions and outlook.
II. MODEL
In this Section we present a general derivation of the
Coulomb matrix elements which couple the multiexciton
configurations differing in the number of electron-hole
pairs. We demonstrate how to include these elements in
the exact diagonalization study of the mixed system and
how to extract physically relevant quantities from that
calculation.
A. Derivation of the coupling Coulomb elements
We start the derivation by writing the all-electron
Hamiltonian for the semiconductor nanostructure. If by
c+i (ci) we denote the creation (annihilation) operator of
an electron on state |i〉, we have:
Hˆ =
∑
i
E˜ic+i ci +
1
2
∑
ijkl
〈ij|Vee|kl〉c+i c+j ckcl, (1)
where E˜i are the single-particle energies of the nanostruc-
ture, while 〈ij|Vee|kl〉 are the Coulomb scattering matrix
elements computed for the single-particle states. Since
in the typical NCs we deal with ∼ 103 atoms, or ∼ 104
electrons, we cannot treat the above Hamiltonian directly
and introduce the language of quasiparticles. To this end,
we divide the basis of single-particle states into two sets:
the valence states, henceforth enumerated by Greek in-
dices, and the conduction states, enumerated with Latin
indices. As a result, the first term of the above Hamil-
tonian will split into two, while the Coulomb operator
will result in the appearance of 24 = 16 terms. Among
these terms there are those which describe the interac-
tion of carriers on conduction states only, as well as that
of carriers on valence states only. Further, we find terms
describing the interaction between conduction and va-
lence carriers, consisting of the direct and exchange com-
ponent. Next, we have terms which describe all possi-
bilities of the Coulomb scattering with transfer of one
carrier from the valence to conduction band or the other
way around. All of them consist of the “direct”-like and
“exchange”-like component. Finally, there are terms de-
scribing the Coulomb scattering with transfer of two car-
riers, from the valence to conduction band and the other
way around. Due to the two-body character of Coulomb
interactions, the above Hamiltonian exhausts all possi-
bilities of Coulomb scattering in the system. We shall
write all these terms explicitely below.
In order to complete the transition into the language
of quasiparticles, we define the “vacuum” reference state
of our system, in which all valence orbitals are occupied,
and all conduction orbitals are empty:
|0〉 =
∏
α
c+α |vac〉, (2)
where |vac〉 denotes the true zero-electron state. The
energy of the state |0〉, E0, is treated as the reference

3level. In what follows, we will consider the charge-neutral
electron-hole excitations from that state, which can be
written as
|i, j, k, . . . , α, β, γ, . . .〉 = c+i c+j c+k . . . h+αh+β h+γ . . . |0〉. (3)
Here the hole creation (annihilation) operators are de-
fined as h+α = cα (hα = c+α ), respectively. As is evident
from the discussion opening this Section, the Hamilto-
nian (1) describes direct coupling of a configuration with
n electron-hole pairs and configurations with n−2, n−1,
n, n + 1, and n + 2 pairs. However, we need to trans-
late it into the language of quasiparticle operators. By
replacing the valence operators with the hole operators
and rearranging terms, we obtain:
HˆQP = HˆCONS + HˆNC , (4)
where the part conserving the number of excitations is
HˆCONS =
∑
i
Eic+i ci +
1
2
∑
ijkl
〈ij|Vee|kl〉c+i c+j ckcl
−
∑
α
Eαh+αhα +
1
2
∑
αβγδ
〈δγ|Vee|βα〉h+αh+β hγhδ
−
∑
iβγl
(〈iγ|Vee|βl〉 − 〈iγ|Vee|lβ〉) c+i h+β hγcl, (5)
and the part changing the number of excitations is
HˆNC =
1
2
∑
ijkδ
(〈ij|Vee|kδ〉 − 〈ij|Vee|δk〉) c+i c+j ckh+δ
+ 12
∑
iβkl
(〈iβ|Vee|kl〉 − 〈βj|Vee|kl〉) c+i hβckcl
+ 12
∑
αβγl
(〈αβ|Vee|γl〉 − 〈αβ|Vee|lγ〉)hαhβh+γ cl
+ 12
∑
αjγδ
(〈αj|Vee|γδ〉 − 〈jα|Vee|γδ〉)hαc+j h+γ h+δ
+ 12
∑
ijγδ
〈ij|Vee|γδ〉c+i c+j h+γ h+δ
+ 12
∑
αβkl
〈αβ|Vee|kl〉hαhβckcl. (6)
In the above Hamiltonian, the single-particle energies are
those of quasiparticles, and therefore have to be properly
dressed in selfenergy and vertex correction terms. We
shall not analyze them in greater detail, as all meth-
ods of calculation of the single-particle structure, i.e.,
k · p, tight-binding, or pseudopotential approaches, at
some point are fitted to the experimental bandgaps, and
therefore are parametrised with already dressed single-
particle energies. Note that all Coulomb elements are
computed with previous, electron rather than the quasi-
particle orbitals. These orbitals are computed directly
in the single-particle methods and care must be taken
to translate the Coulomb elements into the quasiparticle
language. For example, the hole Coulomb matrix ele-
ment 〈αβ|Vhh|γδ〉 = 〈δγ|Vee|βα〉, that is, it is a complex
conjugate of the electron element, while the relation for
electron-hole interactions is even more complicated.
The form of the Hamiltonian (4) allows to appreciate
the terms changing the number of excitations in a clearer
fashion: the terms changing the number of excitations
by one have three quasielectron and one hole operators
(or conversely), while the terms changing the number
of excitations by two are built of four creation or four
annihilation operators.
In semiconductor nanostructures the scattering with
transfer of one carrier must necessarily modify the en-
ergy of a configuration by at least one gap energy Eg,
while the transfer of two carriers introduces a modifica-
tion of at least 2Eg. The largest mixing of configurations
with different numbers of excitations will be seen for con-
figurations close in energy. Therefore, in order to obtain
coupling between a low-lying two-pair excitation, or bi-
exciton (XX), and an exciton (X), one has to consider
highly excited exciton states (excited by at least Eg).6
The X-XX-tri-exciton mixing will start at even higher
energies (of at least 3Eg), i.e., both X and XX must
be highly excited. We shall describe this case here in a
greater detail. Let us first write explicitely the Coulomb
matrix elements coupling the X and XX. We couple a one-
electron-hole-pair configuration |X, iα〉 = c+i h+α |0〉 with a
two-pair configuration |XX, jkβγ〉 = c+j c+k h+β h+γ |0〉 and
obtain:46
〈XX, jkβγ| HˆQP |X, iα〉
= [(〈jk|Vee|βi〉 − 〈jk|Vee|iβ〉) δαγ
+ (〈jk|Vee|iγ〉 − 〈jk|Vee|γi〉) δαβ ]
+ [(〈αj|Vee|γβ〉 − 〈αj|Vee|βγ〉) δik
+ (〈αk|Vee|βγ〉 − 〈αk|Vee|γβ〉) δij ] .(7)
In the above formula, the terms in the first square bracket
apply to the scattering event whereby the creation of an
electron-hole pair is accompanied by a scattering of the
electron constituting the exciton, while the exciton’s hole
must not change its orbital. The second square bracket
contains analogous terms for the case when the exciton’s
hole is scattered, but its electron must stay on the same
orbital. This gives a selection rule for the pair creation
process: the exciton and biexciton configurations must
share at least one carrier. The process in which the
hole is shared is illustrated in the transition Fig. 1(a)-
(b). In Fig. 1(a) we show schematically a one-pair con-
figuration, in which the hole occupies the single-particle
ground state, while the electron is highly excited. In the
scattering process, the electron transfers to a lower single-
particle level, while an additional electron-hole pair is cre-
ated. In Fig. 1(b) this negatively charged trion complex
X− is marked with the rectangle, with the empty cir-
cle denoting the initial level of the excited electron. Note
that the shared hole is a spectator quasiparticle and does
not take part in this process. The scattering event with

4XXX
E
LE
C
TR
O
N
S
TA
TE
S
H
O
LE
S
TA
TE
S
XXX
(a) (b) (c)
h+ X+X- X-
FIG. 1: (Color online) Schematic diagram of the one-pair
excitation (a), the two-pair excitation (b), and the three-pair
excitation (c). Empty circles denote removed quasiparticles.
The negatively charged trion X− appearing as a result of this
scattering event is marked with the rectangles.
shared electron would conform to similar rules, only in
this case we would deal with a highly excited hole con-
verting to a positively charged trion, while the shared
electron would be inert. In the case of the X-XX cou-
pling, the selection rule severely limits the number of
excited X configurations to which any XX configuration
can couple directly. Henceforth we shall refer to these
exciton configurations as X∗1 . However, there are also
excited X configurations, which are not directly coupled
to the XX configuration, but do couple to X∗1 . These con-
figurations shall be referred to as X∗2 . We shall discuss
the importance of X∗2 later on.
Let us now move on to the coupling between the
two- and three-pair excitations, whose one variant is
visualized schematically in the transition Fig. 1(b)-(c).
The matrix element between the two-pair configuration
|XX, jkβγ〉 = c+j c+k h+β h+γ |0〉 and the three-pair config-
uration |XXX, lmnδσπ〉 = c+l c+mc+nh+δ h+σ h+pi |0〉 is com-
puted similarly to the one connecting the one- and two-
pair configurations. We shall not write it here as it is
composed of a large number of terms describing various
permutations of particles. Let us only emphasize that
the scattering process involves a creation of a negatively
charged trion X− out of one excited electron, or a posi-
tively charged trion out of one excited hole, just as in the
case of the X-XX coupling. Figure 1(c) depicts the for-
mer, as the excited electron is scattered down the ladder
of single-particle states (blue arrow) with a simultane-
ous creation of the third electron-hole pair (red arrow).
The resulting negatively charged trion is denoted in the
right-hand panel of Fig. 1(b) by the blue rectangle. The
remaining three “spectator” particles are inert and form
a positively charged trion X+.
Finally, there is also a possibility of coupling the one-
and three-pair exictations directly via the two last terms
of the Hamiltonian (4). However, in this case the energies
of the configurations must necessarily differ by at least
2Eg. This is why in the following we shall neglect this
term.
B. Eigenstates of the mixed system
Presently we solve for the eigenvalues and eigenstates
of the system with a non-constant number of excitations.
To this end, we use a procedure consisting of two steps.
First, we solve for eigenstates and eigenenergies of the
Hamiltonian (4) in the subspaces spanned by the con-
figurations with a conserved number of excitations. In
this case, only the first five terms of that Hamiltonian
are considered. We solve this problem in the exact diag-
onalization approach, in which the Hamiltonian matrix
is written in the basis of configurations of the type given
in Eq. (3) and diagonalized numerically. As a result of
this procedure, for a system with n electron-hole pairs
we obtain the eigenstates of the form
|nX〉p =
∑
i,j,...,αβ...
Ap,ni,j,...,αβ...|i, j, . . . , α, β . . .〉, (8)
where the coefficients Ap,ni,j,...,α,β... compose the p-th eigen-
vector of the Hamiltonian matrix, and the energy of this
state is Ep,n. Since the number of possible n-pair con-
figurations can be very large, we restrict the basis to the
region of energies of interest and control the convergence
of the resulting energy levels with the width of that re-
gion.
In the second step, utilizing the energies and eigen-
states of systems with conserved excitation numbers, we
set up the full matrix of the Hamiltonian (4). The only
nondiagonal elements in this matrix will result from the
Hamiltonian terms changing the number of excitations.
They are computed as linear combinations, which, for
example, for the coupled X∗1 -XX system take the form:
p〈 XX |HˆQP |X∗1 〉q
=
∑
j,k,β,γ
∑
i,α
(
Ap,XXk,l,β,γ
)∗
Aq,Xi,α 〈klβγ|HˆQP |i, α〉. (9)
Note that although the individual coupling elements un-
der the sum may vanish due to the selection rule de-
scribed above, the elements connecting the correlated
states may still be finite. We diagonalize the Hamiltonian
matrix to obtain the eigenstates with mixed number of

5excitations. The K-th state can be written as the linear
combination:
|K〉 = BK |0〉+
∑
i
∑
n
CKi,n|nX〉i. (10)
The energy of this state is referred to as εK .
C. Spectral function and its relation to time
evolution
The degree of mixing between states with different
number of excitations can be extracted from the eigen-
states |K〉 by calculating the spectral function Ap,n(K)
of the p-th state with n electron-hole pairs |nX〉p. This
function is computed as
Ap,n(K) = | p〈 nX |K〉|2 (11)
and, for example, for the p-th biexciton state will take
the form Ap,2(K) = |CKp,2|2, i.e., it is readily obtained
from the eigenvectors of the system. For weak coupling
we expect that the spectral function will have the value
close to 1 for K such that εK ≈ Ep,n, and decay as we
move to the other eigenstates |K〉.
In practice, one is typically interested in the lifetime of
the multiexciton state and attempts to engineer the sys-
tem so as to achieve long lifetimes of states with many
excitons. If the dynamics of the system is governed by the
Hamiltonian (4) only and is not changed by any incoher-
ent relaxation processes, we can trace the time evolution
of the state |Ψ〉 of the coupled system simply by
|Ψ(t)〉 = exp
(
− ih¯ HˆQP t
)
|nX〉p, (12)
assuming that the system is prepared in the state |nX〉p.
Since |nX〉p =
∑
K
(
CKp,n
)∗ |K〉, we have
|Ψ(t)〉 =
∑
K
exp
(
− ih¯εKt
)
(
CKp,n
)∗ |K〉, (13)
and we can observe the time evolution
of our state by computing the projection
| p〈 nX |Ψ(t)〉|2 =
∣
∣
∣
∑
K exp
(
− ih¯εKt
) ∣
∣CKp,n
∣
∣
2
∣
∣
∣
2
=
∣
∣
∑
K exp
(
− ih¯εKt
)
Ap,n(K)
∣
∣
2. The time evolution of
the system, and in particular the change of the number
of excitations from the one prepared in the system,
is related to the Fourier transform of the spectral
function in the time domain. At this point it is useful to
consider two limiting examples of the spectral function.
First, if Ap,n(K) is finite only for several states K, we
may expect a complex time evolution, with oscillating
contributions from all these states.
Second, if our state |nX〉p is immersed in a quasi-
continuous spectrum of other states (possibly with differ-
ent n) and if its spectral function can be approximated by
a Lorentzian, Ap,n(K) ≈ (γ/2π)/[(εK −Ep,n)2 + (γ/2)2]
and Ap,n(K) = A0 on resonance, then the time evolution
is described by an exponential decay, | p〈 nX |Ψ(t)〉|2 =
|A0 +(1−A0) exp(−γt/2)|2, with γ being the character-
istic width of the spectral function. At long times and
with strong mixing (the value of A0 ≪ 1), the state of the
system can no longer be identified with the state |nX〉p,
as the probability density is distributed coherently in the
multitude of states of the system. The above analysis
shows that the characteristic decay time constant 1/γ
is not related simply to the bare coupling between the
state |nX〉p and other states at the same energy, as this
will predominantly affect the spectral function maximum
value A0. The dynamics of this “dissolution” of our state
|nX〉p is decided by its coupling to states off-resonance:
the stronger that coupling, the broader the spectral func-
tion and the faster the decay. Note also that the proba-
bility of finding the system in the state |nX〉p does not
decay to zero, but rather the amplitude A20. We shall
supplement this intuitive picture with a detailed analysis
of the dynamics in the next Section.
III. EXCITON-BIEXCITON COUPLING IN
CDSE NANOCRYSTALS
In this Section we apply the general approach to de-
scribing the dynamics of the bi-exciton XX in a CdSe
nanocrystal with diameter of 3.8 nm, as studied in
Ref. 47. We choose to focus on the low-energy XX states,
which allows us to consider their coupling only to excited
exciton states of similar energy. Below we will discuss the
entire procedure, starting from the computation of single-
particle states, then the correlated states of X and XX,
coupling matrix elements, treatment of the mixed system
and extraction and analysis of the spectral function.
A. Single-particle states
We start the parametrization of the Hamiltonian (4)
with the computation of single-particle states and their
energies. To this end we utilize the atomistic tight-
binding sp3d5s∗ approach. We look for the single-particle
wave function |i〉 in the form of a linear combination
|i〉 =
∑
R,α
F (i)R,α|R,α〉 (14)
of atomic orbitals |R,α〉. The index α enumerates the
types of orbitals (s, three p, five d, and s∗, of which all
are degenerate spin-doublets), while the index R enumer-
ates atoms. The coefficients F (i)R,α and the energies Ei of
states are computed by diagonalizing the tight-binding
Hamiltonian
HˆTB =
∑
R,α
εR,αc+R,αcR,α +
∑
R,α,β
λSOR,α,βc+R,αcR,β